Homoclinic bifurcations in a neutral delay model of a transmission line oscillator

In a transmission line oscillator (TLO) a linear wave travels along a piece of cable, the transmission line, and interacts with terminating electrical components. A fixed time delay arises due to the transmission time through the transmission line. Recent experiments on a TLO driven by a negative resistor demonstrated rich delay-induced dynamics and high-frequency chaotic behaviour. Furthermore, good agreement was found with a neutral delay differential equation (NDDE) model. In this paper we perform a numerical bifurcation analysis of the NDDE model of the TLO. Our main focus is on homoclinic orbits, which give rise to complicated dynamics and bifurcations. For small time delay there is a homoclinic orbit to a steady-state. However, past a codimension-two Shil'nikov-Hopf bifurcation the homoclinic orbit connects to a saddle-type periodic solution, which exists in a region bounded by homoclinic tangencies. Both types of homoclinic bifurcations are associated with accumulating branches of periodic solutions. We summarize our results in a two-parameter bifurcation diagram in the plane of resistance against time delay. Our study demonstrates that the theory of homoclinic bifurcations in ordinary differential equations largely carries over to NDDEs. However, we find that the neutral delay nature of the problem influences some bifurcations, especially convergence rates of folds associated with the homoclinic tangencies.